Born Rule

In one sentence: The Born rule says the chance of finding a particle at a spot, or of getting an outcome, is the squared size of its wave at that spot.

$P(x) = |\psi(x)|^2$

Read it out loud: “P of x equals the absolute value of psi of x, squared. P of x is the probability density at position x, and psi is the wave function.”

What each symbol means

Symbol	Meaning
$P(x)$	The probability density at position $x$ : probability per unit length, so per meter in one dimension. To get a real probability, multiply by a width and add up over a region.
$x$	A position. In one dimension it is measured in meters (m).
$\psi(x)$	The wave function at position $x$ . It is a complex number, not a probability. In one dimension its units are $\text{m}^{-1/2}$ .
$\|\psi(x)\|^2$	The squared size of the wave function. The size (absolute value) removes the phase, and squaring makes the result positive.
$c_i$	The amplitude (a complex number) for possible outcome number $i$ , in a state written as a sum of possible outcomes.
$P_i$	The probability of outcome number $i$ : a number between 0 and 1.

What the formula is doing

The wave function $\psi$ describes the state of a particle, but $\psi$ itself is a complex number, so it cannot be a probability. In 1926 Max Born proposed that the probability is given by the squared size of $\psi$ . For a particle in one dimension, $P(x) = |\psi(x)|^2$ is a probability density: the probability per unit length near position $x$ .

A density is not a probability by itself. To get the chance that the particle is in a region, add up $P(x)$ over that region, which is what an integral does. Over all of space the total must be exactly 1, because the particle is somewhere.

For a set of possible outcomes the rule is simpler. Suppose the state has one amplitude $c_i$ for each possible outcome $i$ . Then $P_i = |c_i|^2$ : take the size of the amplitude and square it. The amplitudes must be chosen so that $\sum_i |c_i|^2 = 1$ .

Amplitudes are complex, so each one has a direction (its phase) as well as a size. The probability uses only the size, but the phase decides how different possibilities add or cancel when they combine. That cancellation is behind interference patterns.

The rule does not say which outcome you will get. It gives only the probabilities. One measurement gives one outcome, chosen at random with those probabilities, and repeating the experiment on identical states reveals the pattern.

An everyday picture

Think of a crowd spread over a park. The number of people per square meter is the density, and the spot with the most people is the spot where you are most likely to find someone. The Born rule sets the crowd density at each spot to the squared size of the wave there. The wave itself can dip below zero or point in different directions, but its squared size is always positive, as a crowd count always is.

Worked example: A qubit with two possible outcomes

A qubit is in a state with amplitudes $c_0 = 0.6$ for outcome 0 and $c_1 = 0.8i$ for outcome 1. Here $i$ is the imaginary unit, with $i^2 = -1$ .

Size of the second amplitude: $|c_1|^2 = |0.8i|^2 = 0.8^2 = 0.64$ .
Size of the first amplitude: $|c_0|^2 = 0.6^2 = 0.36$ .
Apply the rule: $P(0) = |c_0|^2 = 0.36$ and $P(1) = |c_1|^2 = 0.64$ .
Check the total: $0.36 + 0.64 = 1$ , so the two outcomes cover every possibility.

Answer: Outcome 0 has probability 0.36 (36%) and outcome 1 has probability 0.64 (64%). The $i$ in $c_1$ does not change these numbers, because only the size of $c_1$ is squared. The phase would matter if the state were measured in a different way, where the two amplitudes could combine.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Squaring the amplitude instead of its size. For a complex number, $|c|^2$ is not $c^2$ . For $c = 0.8i$ , $c^2 = -0.64$ , which is negative and cannot be a probability.
Treating $\psi$ itself as the probability. $\psi$ can be negative or complex. Only $|\psi|^2$ is a probability density.
Reading $P(x)$ as a chance at one exact point. For continuous position, the chance of exactly one point is zero. You need a small interval, so multiply the density by its width or add it up over a region.
Forgetting that the probabilities must total 1. If they do not, divide each amplitude by the square root of the total probability to normalize the state.

Where you will meet it

Chemistry, where the squared wave function of an electron gives the electron density shown in orbital pictures.
Quantum computing, where the chance of each measurement result on a qubit comes from squared amplitudes.
Particle physics, where the probability of a scattering process is computed from the square of its amplitude.
Quantum tunneling, where the chance of finding a particle beyond a barrier is computed from the squared wave function.

Questions people ask

Why square the size instead of using it directly?

Squaring always gives a positive number and works for complex amplitudes. In the standard formulation of quantum mechanics the rule is a basic postulate, and experiments have checked it in many settings.

Does measuring collapse the wave function?

Standard quantum mechanics says a measurement gives one outcome, chosen with the probabilities above, and the state afterward matches that outcome. What exactly happens during a measurement is an open question that the interpretations of quantum mechanics answer differently.

Why must the probabilities add up to 1?

The particle must be found somewhere, so one of the listed outcomes must happen. The total chance of all outcomes is 100%, which is written as 1.

Schrödinger Equation: The Schrödinger equation says how a quantum particle’s wave changes over time, the way Newton’s laws say how a moving ball changes its position.
Heisenberg Uncertainty Principle: The more tightly a particle’s position is pinned down, the more its speed must spread out, and no instrument can beat that trade-off.
De Broglie Wavelength: Every moving object has a wavelength, and the heavier or faster it moves, the shorter that wavelength gets.